An Equilibrium Analysis of the Long-Term Care Insurance Market

Transcription

1 An Equilibrium Analysis of the Long-Term Care Insurance Market Ami Ko First version: November 2016 This version: April 2018 Abstract A life-cycle model of intergenerational long-term care decisions is developed to analyze how family interactions affect the equilibrium coverage and welfare in the U.S. long-term care insurance market. In the model, consistent with empirical facts, help provided by adult children substitutes formal long-term care services, and parents purchase of long-term care insurance discourages children s informal care provision. The model is structurally estimated using data from the Health and Retirement Study by the conditional choice probability (CCP) estimation method. I find that wealthy parents have a disincentive to purchase insurance as it reduces their children s strategic incentive to provide informal care. I also find that the failure to account for heterogeneity in informal care options results in adverse selection where the long-term care insurance market attracts a disproportionate number of individuals with worse informal care options and hence higher expected formal care spending. I demonstrate that using family demographics in pricing long-term care insurance contracts reduces adverse selection and improves the average welfare of the elderly. Georgetown University, ICC 557, Department of Economics, 37th and O Streets, N.W., Washington D.C ami.ko@georgetown.edu. I am extremely grateful to my advisors, Hanming Fang, Holger Sieg, and Petra Todd for their guidance and support throughout this project. I have benefited greatly from discussions with Naoki Aizawa, Daniel Barczyk, Mary Ann Bronson, Norma Coe, George-Levi Gayle, Qing Gong, Nick Janetos, Matthias Kredler, Rasmus Lentz, Lee Lockwood, Costas Meghir, Robert Miller, Giuseppe Moscarini, John Rust, Andrew Shephard, Daniel Silverman, Dongho Song, Jan Tilly, Gustavo Ventura, and Kenneth Wolpin. I also thank seminar participants at Arizona State University, Boston College, Georgetown University, University of Pennsylvania, University of Rochester, UCL Structural Dynamic Conference, AEA Meetings, ASHEcon, CRR at Boston College, Econometric Society North American Summer Meetings, and Social Security Administration RRC Annual Meetings for helpful comments and discussions. I gratefully acknowledge financial support from the Korea Foundation for Advanced Studies and the Center for Retirement Research at Boston College (BC Grant ) sponsored by the Social Security Administration. 1

2 1 Introduction Elderly individuals face substantial risks of having functional limitations and hence requiring longterm care. In the U.S., about three fourths of 60-year-olds will have chronic conditions resulting in daily activity limitations, while the other fourth will have no such conditions until death. Formal long-term care services are expensive with the median annual cost for nursing homes exceeding $90,000. Public insurance, Medicaid, exists, but is means-tested and one has to be impoverished to be eligible for benefits. Yet, only about 10 percent of the elderly own private long-term care insurance which insures these large formal long-term care expenditure risks. Many researchers have studied why this market is so small, and have identified Medicaid (Brown and Finkelstein, 2008), bequest motives (Lockwood, 2016), private information (Hendren, 2013), market power and administrative costs (Braun, Kopecky, and Koreshkova, 2017) as relevant factors. However, scant attention has been given to the role of unpaid care provided by the family, usually adult children, as a substitute to formal long-term care. In this paper, I develop and structurally estimate an intergenerational game featuring elderly parents long-term care insurance choice, formal care utilization, and savings, and adult children s informal care provision and labor supply. I embed the estimated intergenerational game within an equilibrium long-term care insurance market to quantify the equilibrium effects of family interactions and explore welfare-increasing policies. The model incorporates two key mechanisms. First, it incorporates the possibility of adverse selection due to asymmetric information about children s informal care giving. As the provision of long-term care, i.e., assistance with daily tasks, does not require much professional training, informal care provided by adult children substitutes formal care services (Houtven and Norton, 2004; Charles and Sevak, 2005; Houtven and Norton, 2008; Coe, Goda, and Van Houtven, 2015). Even among individuals with public or private long-term care insurance, almost one half rely on informal care from children. 1 As long-term care insurance companies pay only for formal longterm care services, whether a consumer has children who are likely to provide informal care is highly relevant for insurance companies costs. However, no long-term care insurance company uses information about children in pricing insurance contracts, despite the absence of any regulation. This suggests that insurance companies could attract a disproportionate number of individuals who have worse informal care options and therefore have higher expected formal care spending. To quantify the welfare cost of such adverse selection, I develop a model in which elderly individuals face heterogeneous informal care likelihood from children and make insurance decisions based on that dimension of heterogeneity. In this regard, this paper is related to the vast empirical literature on asymmetric information in insurance markets (see Einav, Finkelstein, and Levin (2010) for a survey of the literature). My contribution lies in identifying the availability of a close substitute, i.e., informal care, as the main source of private information. 1 Author s calculation using data from the Health and Retirement Study. The sample is restricted to elderly individuals with daily activity limitations who have either Medicaid or private long-term care insurance. 2

3 Second, the model incorporates the possibility that children strategically provide informal care to prevent depletion of parent savings on expensive formal care services, and knowing this, elderly parents refrain from purchasing long-term care insurance. This strategic non-purchase of insurance has long been argued by theoretical studies such as Bernheim, Shleifer, and Summers (1985), Pauly (1990), Zweifel and Struwe (1996), and Courbage and Zweifel (2011). Existing reducedform evidence in the literature also suggests that strategic considerations may be important. For example, Brown (2006) and Groneck (2016) find that parents reward informal caregiving children with a much higher bequest compared to non-caregiving children. By developing and estimating an intergenerational game over long-term care decisions, I provide the first estimate for the effects of strategic interactions of the family on the insurance market equilibrium. This is in contrast to the recent work by Mommaerts (2015) who uses a cooperative model to study long-term care decisions of the family, and analyzes only the demand for long-term care insurance. Within the dynamic intergenerational game that I develop, an elderly parent and an adult child interact non-cooperatively from the parent s retirement to death. 2 The parent makes a once-andfor-all long-term care insurance purchase decision at retirement, and from then on, experiences health shocks and makes decisions concerning formal care utilization and savings, with Medicaid incorporated as means-tested public long-term care insurance. In this regard, this paper contributes to the literature on elderly savings and medical expenditure uncertainty such as Hubbard, Skinner, and Zeldes (1995), Palumbo (1999), and De Nardi, French, and Jones (2010). The adult child in each period allocates time to informal care provision, work and leisure over the parent s life-cycle. 3 The child may provide informal care out of altruism or to prevent depletion of her inheritance on formal care. The child s strategic incentive to provide help is diminished if the parent has long-term care insurance, because in that case, insurance companies cover the formal care costs. The parent is forward looking, so if the parent prefers informal care to formal care, she may demand less insurance to avoid diminishing the child s strategic informal care incentive. This is how the model incorporates strategic non-purchase of insurance. Furthermore, the richness of child and parent level heterogeneity results in heterogeneous informal care likelihood across families, allowing for the analysis of insurance selection based on this dimension of private information. Owing to the finite horizon assumption with sequential moves, for a given price of long-term care insurance, the intergenerational game has a unique equilibrium. I estimate the intergenerational game using the Health and Retirement Study (HRS) and actual premium data over the sample period. The estimation entails a large computational cost because the model is a dynamic game with many state variables and rich individual heterogeneity. I address the computational challenge by employing a two-stage conditional choice probability (CCP) estimator pioneered by Hotz and Miller (1993) and Hotz, Miller, Sanders, and Smith (1994), and 2 This feature of the model is related to Barczyk and Kredler (forthcoming) and Fahle (2014) who also estimate a dynamic game of intergenerational care arrangements, but in contrast to this paper, do not incorporate insurance purchase decisions. 3 Skira (2015) also estimates a dynamic model of an adult child s informal care and work choices, but abstracts away from intergenerational interactions. 3

4 further developed in the context of dynamic games by Aguirregabiria and Mira (2007), Bajari, Benkard, and Levin (2007), Pakes, Ostrovsky, and Berry (2007) and Pesendorfer and Schmidt- Dengler (2008). The main idea behind CCP estimators is that it is not necessary to fully solve a dynamic model in order to obtain value functions. Once nonparametric estimates of conditional choice probabilities are obtained in the first stage, one can use these probability estimates to obtain nonparametric estimates of value functions. I use forward simulation to construct value functions in the second stage as in Bajari, Benkard, and Levin (2007). 4 While mostly used in the industrial organization literature, this paper is the first to apply the CCP estimator in estimating a dynamic game of intergenerational decisions. I recover parents preferences for leaving bequests and receiving informal or formal care, as well as children s preferences for leisure, consumption, and providing informal care. The estimated model is capable of matching the most important features of the data including the monotonically increasing long-term care insurance ownership rate in wealth and the inverted-u pattern of informal care receipt across wealth. To embed the estimated intergenerational game within an equilibrium long-term care insurance market, I incorporate perfectly competitive risk-neutral insurance companies. The market equilibrium is characterized by the break-even price that leads to zero profits. As the estimated intergenerational game predicts, for a given price of long-term care insurance, the insurance demand and the expected formal care spending which are the main inputs in calculating insurers profits, I iteratively solve the game until I find the break-even price. Using the equilibrium insurance market framework, I conduct counterfactuals and obtain the following results. First, there is quantitatively meaningful strategic non-purchase of insurance. When children do not strategically reduce informal care provision in response to parents purchase of long-term care insurance, the equilibrium ownership rate increases by over 7 percentage points, corresponding to a 63 percent increase. This effect is the greatest among wealthy parents as their children have the strongest strategic incentive to provide care. Second, the failure to account for heterogeneity in informal care options results in adverse selection. Compared to parents who almost always receive informal care in the event of bad health shocks, parents who almost never do demand insurance twice as much and have expected lifetime formal care spending that is higher by almost $100,000. Third, pricing insurance contracts based on family characteristics that highly predict informal care provision from children, such as the presence of a daughter and the number of children, reduces the amounts of private information about informal care options and increases the average welfare of the elderly by almost $5,500. Finally, as an application of my model, I provide potential reasons for the recent premium increases in the long-term care insurance market. Starting around 2013, which is after the sample period of the estimation data, there have been sharp premium increases not just for new policies but also for existing policies. Rate increases on existing policies were approved by the state gov- 4 By specifying flow utilities that are linear in structural parameters, the forward simulation procedure is performed only once which greatly reduces the computational cost. 4

5 ernments on the basis that policies had been substantially underpriced. 5 My model, which allows rich interactions between insurance demand, formal care utilization, and informal care provision, finds that premiums of typical contracts in 2002 were below the break-even premium by almost 80 percent. The number is consistent with the requested premium increases of percent from major long-term care insurance companies on their older blocks of policies. This result serves as external validation of the model. I discuss potential reasons for underpricing and the timing of the premium hikes. The rest of this paper proceeds as follows. Section 2 presents empirical facts about long-term care in the U.S. Section 3 presents the model. Section 4 presents the data and the estimation results. Section 5 presents the main results. Section 6 concludes. 2 Empirical Facts I start by providing empirical facts about the U.S. long-term care sector. The main data for this paper come from the Health and Retirement Study (HRS), which surveys a representative sample of Americans over the age of 50 every two years since Using seven interviews from the HRS , I provide empirical patterns that motivate the model of intergenerational long-term care decisions presented in the next section. 2.1 Background Substantial long-term care risk. Long-term care is formally defined as assistance with basic personal tasks of everyday life, called Activities of Daily Living (ADLs) or Instrumental Activities of Daily Living (IADLs). Examples of ADLs include bathing, dressing, using the toilet, and getting in and out of bed. IADLs refer to activities that require more skills than ADLs such as doing housework, managing money, using the telephone, and taking medication. Declines in physical or mental abilities are the main reasons for requiring long-term care. Using individuals aged 60 and over in the HRS , Figure 1 reports, for each gender and age group, the share of individuals who have ADL/IADL limitations or are cognitively impaired. Long-term care needs rise sharply with age and over 60 percent of individuals aged 85 and older need assistance with daily tasks. However, not everybody develops ADL/IADL limitations towards the end of their lives. In fact, about 32 percent (19 percent) of healthy 60-year-old men (women) will never need long-term care until their death. 6 These findings suggest that elderly individuals face substantial risks about how much long-term care they would need. 5 This is in contrast to the existing findings in the literature. Brown and Finkelstein (2007) use an actuarial model of formal long-term care utilization probabilities, and find the average markup of 18 percent for policies sold in Author s calculation using the HRS I provide details about the estimation in Section

6 Figure 1: Long-Term Care (LTC) Needs by Age Men Women Notes: The figure reports the share of respondents who have ADL/IADL limitations or are in the bottom 10 percent of the cognitive score distribution. The sample is limited to individuals aged 60 and over in the HRS Informal care as the backbone of long-term care delivery. Unpaid long-term care provided by the family - which I refer to as informal care in this paper - plays a substantial role in the long-term care sector. This is because unlike acute medical care, long-term care does not require professional training; it simply refers to assistance with basic personal tasks. Several studies have documented the importance of informal care in the U.S. long-term care sector. For example, work by Barczyk and Kredler (forthcoming) shows that informal care accounts for 64 percent of all help hours received by the elderly. Using the HRS , I find that 62 percent of individuals with long-term care needs receive some help from children, and among these individuals, almost 80 percent report just one child as the primary caregiver. Table 1 reports how family characteristics vary between individuals who receive informal care from children and those who do not. The number of children, and the presence of a daughter and a child residing in proximity appear to have strong positive correlations with the chance of receiving informal care. Only 4 percent of parents pay their children for help, implying that immediate financial compensation for informal care is rare. Costly formal care services. Another way to meet one s long-term care needs is to use formal long-term care services, such as nursing homes, assisted living facilities, and paid home care. These formal care services are labor-intensive and expensive. In 2017, the median annual rate was $97,000 for a private room in a nursing home, $45,000 for assisted living facilities, and $48,000 for paid home care. 7 Combined with substantial risks of needing long-term care, formal care is one of the 7 Source: Genworth. 6

7 Table 1: Informal Care Receipt and Family Characteristics Sample receiving Sample not receiving informal care informal care Average number of children Share who have a daughter Share who have a child within a 10 mile radius Share who pay children for help Observations 3,464 2,123 Notes: Sample size = 5,587. The sample is limited to single respondents aged 60 and over who have daily activity limitations. largest financial risks faced by the elderly; 40 percent of 65-year-olds will not have any formal care expenses, while 60 percent will incur on average $100,000 during their remaining life and 5 percent will spend more than $300,000 in 2017 dollars (Kemper, Komisar, and Alecxih, 2005/2006). A very small long-term care insurance market. Private long-term care insurance provides financial protection against these large formal care risks. The U.S. long-term care insurance market is relatively young and modern insurance products were introduced in the late 1980s. 8 Typical longterm care insurance contracts cover both facility care and paid home care provided by employees of home care agencies. Most do not cover informal care. For underwriting purposes, insurance companies perform cognitive tests, and require medical records, blood and urine samples. information about applicants children, who are most likely to be primary informal caregivers, is collected. Premiums are conditional on age, gender, and underwriting class determined by health conditions. Gender-based pricing was newly adopted in 2013, despite the well-known fact that women have a higher chance of using formal care compared to men; the probability of ever using formal care is 40 percent for 65-year-old men, and 54 percent for 65-year-old women (Brown and Finkelstein, 2008). Contracts are guaranteed renewable and specify a constant and nominal annual premium. The average purchase age is 60 years, but most people do not use insurance until they turn 80 (Broker World, ). Using the HRS , I find that only 13 percent among individuals aged 60 and over own some form of private long-term care insurance. Medicaid as the biggest payer for formal care services. No Based on a recent report by the Kaiser Family Foundation, formal long-term care expenses totaled over $310 billion in 2013, which is close to 2 percent of GDP. 9 Medicaid is the biggest payer, accounting for 51 percent of the total payments, followed by other public insurance programs (21 percent), out-of-pocket (19 percent), and private long-term care insurance (8 percent). In contrast to a common misconception, Medicare coverage for long-term care is very limited. Only nursing home stays following a qualified hospital stay are covered up to 100 days, and there are substantial copayments for days Medicaid, on 8 Source: National Care Planning Council. 9 The report can be found at 7

8 the other hand, provides unlimited coverage to eligible individuals with limited assets. While one has to be almost impoverished to be eligible for benefits, individuals can spend-down their assets until they meet Medicaid eligibility requirements, which has been identified as an important factor in explaining the limited size of the long-term care insurance market (Brown and Finkelstein, 2008). At $158 billion in 2013, Medicaid spending on long-term care imposes severe fiscal constraints at both state and federal government levels (Commission on Long-Term Care, 2013). 2.2 Private Information in the Long-Term Care Insurance Market Despite the fact that informal care plays a critical role in delivering long-term care, long-term care insurance companies do not collect any information about children from consumers. This is not because of regulation as there are no restrictions on the characteristics that may be used in pricing long-term care insurance contracts (Brown and Finkelstein, 2007). I now provide evidence that private information about children s informal care provision may be an important source of adverse selection in the long-term care insurance market. To measure private information about the availability of informal care, I use responses to the following HRS question: Suppose in the future, you needed help with basic personal care activities like eating or dressing. Will your daughter/son be willing and able to help you over a long period of time? If beliefs about children s expected informal care provision were an important dimension of private information in the long-term care insurance market, then conditional on information used by insurance companies in setting prices, the responses to this question would be correlated with the formal care risk and the insurance demand in a statistically significant way. To examine if this were true, following the empirical strategy in Finkelstein and McGarry (2006), I estimate the following probit equations: P r(nh i,t t+5 = 1) = Φ(α 1 B IC it P r(lt CI it = 1) = Φ(α 2 B IC it + β 1 B NH it + X it γ 1 ) and (1) + β 2 B NH it + X it γ 2 ). (2) The term NH i,t t+5 is an indicator for staying in a nursing home for more than 100 nights in the next five years since the interview. 10 holdings. LT CI it is an indicator for current long-term care insurance X it is a vector of individual characteristics used by insurance companies for pricing that includes age, gender, and various health conditions. 11 It does not include any information about children as such information is not collected by insurance companies. Bit IC is an indicator for whether the individual thinks children will provide informal care. Bit NH is the individual s selfassessed probability of entering a nursing home in the next five years. I include this term to compare 10 Short-term nursing home stays following acute hospitalization are covered by Medicare up to 100 days. To distinguish nursing home stays that are covered by private long-term care insurance from those covered by Medicare, I use nursing home stays lasting more than 100 nights. 11 I follow Finkelstein and McGarry (2006) and Hendren (2013) to control for pricing covariates. 8

9 Table 2: Beliefs about Informal Care, Nursing Home Use, and Insurance Coverage (1) (2) Believe children Not believe children will help will help Subsequent NH Use LTCI Observations 2,553 2,552 Notes: Column (1) reports the nursing home (NH) utilization rate over the next five years and the current long-term care insurance (LTCI) coverage rate of respondents who believe their children will help with longterm care needs in the future. Column (2) reports the subsequent nursing home utilization and insurance coverage rates of respondents who do not believe their children will help. The sample is limited to individuals aged who have children and do not have rejection conditions based on underwriting guidelines in Hendren (2013). the importance of private information about informal care options (Bit IC ) to other dimensions of private information, such as unobserved health, that may be captured in Bit NH. 12 To estimate the probit equations, I restrict the sample to individuals who are healthy enough to buy long-term care insurance at the time of interview, and old enough to have long-term care needs over the next five years since the interview. I use individuals aged who have children and do not have conditions that render them ineligible to buy long-term care insurance. 13 Table 2 reports the subsequent nursing home utilization rate and the long-term care insurance coverage rate of the sample broken down by whether the respondent believes children will help. About one half of the sample believes children will help. These beliefs appear reasonable because in the data, about 60 percent of respondents with long-term care needs actually receive care from their children. Individuals who believe children will help are less likely to enter a nursing home in the future and to own long-term care insurance. Table 3 reports the results from the probit estimation. Column (1) shows that individual beliefs about children s informal care provision are powerful predictors of subsequent nursing home use. Individuals who believe their children will help are 1 percentage point less likely to enter a nursing home in the future. This is a substantial effect as 2 percent of the sample use nursing homes in the next five years. 14 What is surprising is that individual beliefs about nursing home entry have no power in predicting subsequent nursing home use; the relationship is indeed negative and statistically insignificant. 15 If beliefs about nursing home entry reflect information about unobserved 12 Several studies have used self-assessed probability of entering a nursing home, Bit NH, to construct a measure of private information about formal care risk (Finkelstein and McGarry, 2006; Hendren, 2013). 13 I follow Hendren (2013) to identify rejection conditions. I exclude individuals who have ADL/IADL limitations, have experienced a stroke, or have used nursing homes or paid home care in the past. 14 The negative and significant correlation between beliefs about informal care and subsequent nursing home use holds true when I measure nursing home use over a longer time horizon. 15 This result is consistent with Hendren (2013) who finds little predictive power of beliefs about nursing home entry among individuals who are eligible to buy long-term care insurance. The fact that beliefs about informal care provision by children have predictive power, while beliefs about nursing home entry do not, suggests individuals imperfect ability to incorporate all relevant information in forming these beliefs. As 9

10 Table 3: Results from the Asymmetric Information Test (1) (2) Subsequent NH use LTCI Believe children will help (0.004) (0.012) Subjective prob of future NH use (0-1) (0.012) (0.029) Female (0.157) (0.390) Age (0.002) (0.004) Female*Age (0.002) (0.005) Psychological condition (0.007) (0.024) Diabetes (0.005) (0.019) Lung disease (0.007) (0.025) Arthritis (0.004) (0.013) Heart disease (0.005) (0.017) Cancer (0.006) (0.018) High blood pressure (0.004) (0.014) Cognitive score (0-1) (0.020) (0.050) Observations 5,105 5,105 Notes: p < 0.10, p < 0.05, p < Reported coefficients are marginal effects from probit estimation of Equations (1) and (2). Standard errors are clustered at the household level and are reported in parentheses. Dependent variable in Column (1) is an indicator for staying in a nursing home for more than 100 nights in the next five years. Mean is Dependent variable in Column (2) is an indicator for long-term care insurance ownership. Mean is The sample is limited to individuals aged who have children and do not have rejection conditions based on underwriting guidelines in Hendren (2013). health conditions, the insignificant relationship suggests that the amounts of private information about health are small. Column (2) indicates that there is a negative and significant relationship between beliefs about children s informal care provision and insurance holdings. Individuals who believe their children will help are 4 percentage points less likely to own long-term care insurance. Given the coverage rate of 16 percent among the sample, this finding serves as suggestive evidence that beliefs about children s expected informal care provision have a substantial effect on insurance choices. Taken together, Table 3 provides suggestive evidence that (1) the dimension of private information that is the most relevant to long-term care insurance companies is individuals beliefs about children s informal care provision, and (2) individuals with worse informal care options and hence higher expected formal care spending may be more likely to select into insurance, creating potential adverse selection. argued in Finkelstein and McGarry (2006), if B NH it nursing home use, then conditional on Bit NH no power in predicting nursing home use. is a sufficient statistic for private information about, all other individual information (including Bit IC ) should have 10

11 Figure 2: Long-Term Care Insurance Coverage and Medicaid Eligibility by Wealth Notes: The black solid line represents the long-term care insurance coverage rate by wealth quintile. The dashed line represents the share of respondents on Medicaid. The gray solid line represents the share of respondents who have either long-term care insurance or Medicaid benefits. The sample is limited to single respondents aged 60 and over in the HRS Strategic Informal Care and Bequests Several theoretical papers like Bernheim, Shleifer, and Summers (1985), Pauly (1990), Zweifel and Struwe (1996), and Courbage and Zweifel (2011) have long argued that parents may forgo insurance because with insurance, parents cannot use bequests as an effective instrument to elicit favorable behaviors from children (for example, informal care provision). I now provide descriptive statistics that suggest that bequests may play an important role in shaping children s informal care decisions. Given the costly nature of formal care, children may provide care themselves to protect bequests from formal care expenses. If that is the case, the out-of-pocket costs of formal care that parents face may be an important factor in children s informal care decisions. For example, if parents face zero out-of-pocket costs of formal care by having full long-term care insurance or being Medicaid eligible, children will not have any strategic incentive to provide informal care. Based on this intuition, I look for data patterns that suggest a positive relationship between children s informal care provision and parents out-of-pocket costs of formal care. Figure 2 reports the long-term care insurance coverage rate (solid line) and the share of Medicaid eligibles (dashed line) by wealth quintile. The long-term care insurance coverage rate increases in wealth while the share of Medicaid eligibles decreases in wealth. Individuals in the middle of the wealth distribution face the largest out-of-pocket costs of formal care as the share covered by either long-term care insurance or Medicaid is the lowest. Indeed, Figure 3 shows that there is an inverted- U pattern of informal care receipt; middle-wealth parents receive the most informal care from children at the extensive and intensive margins. While other factors, such as children s opportunity 11

12 Figure 3: Informal Care from Children by Parent Wealth Notes: The left panel reports the share of respondents receiving care from children by respondent wealth quintile. The right panel reports the average monthly care hours provided by children. The sample is limited to single respondents aged 60 and over who have long-term care needs in the HRS costs, may contribute to the inverted-u pattern of informal care, the positive relationship between children s informal care provision and parents out-of-pocket costs of formal care suggest that children may provide informal care to protect bequests from formal care expenses. In Appendix A, I present further descriptive evidence that long-term care insurance undermines children s informal care incentives. Several empirical studies also find a significant relationship between bequests and children s informal care behaviors. Brown (2006) uses inclusion in life insurance policies and wills as proxies for bequests and finds that caregiving children are more likely to receive end-of-life transfers from parents. Groneck (2016) uses actual bequest data obtained from the HRS exit interviews and finds that caregiving children, on average, receive bequest amounts that are twice as much as those received by non-caregiving children. These findings provide further support for strategicallymotivated informal care and bequests. 3 Intergenerational Game This section presents the model that I develop to derive the demand for long-term care insurance, which incorporates the possibility that (1) elderly parents make long-term care insurance purchase decisions with beliefs over children s informal care decisions, and (2) children provide informal care in part to prevent depletion of parental savings on formal care costs. For now, I abstract away from the supply side of the long-term care insurance market and assume standard long term care 12

13 insurance policies are sold at a given price. I explicitly introduce the supply side and define the insurance market equilibrium in Section 5 where I present counterfactuals. The model presented in this section describes interactions between an elderly parent and an adult child from the parent s retirement to death. 16 To incorporate strategic concerns, the model assumes the parent and the child interact non-cooperatively, each making decisions to maximize his or her own expected utility with beliefs over the other player s actions. In each period, the child makes a labor force participation decision, and when the parent experiences a bad health shock, the child makes an informal care supply decision. As the child inherits the parent s wealth, the child may provide informal care to protect parental savings from formal care expenses. The parent makes a once-and-for-all long-term care insurance purchase decision at retirement, and from then on, experiences health shocks, and makes decisions concerning formal care utilization, and savings. As the parent does not use formal care while receiving informal care, the parent s demand for long-term care insurance depends on her beliefs about the child s informal care decisions. The model incorporates Medicaid as means-tested public long-term care insurance, and rich parent and child level heterogeneity. 3.1 Environment Time, indexed by t, is discrete and finite. Variables related to the parent will have superscript P, and variables related to the child will have superscript K. This is a model of one parent and one child, so I omit i subscripts. Let age P t and age K t denote the parent s and the child s age in period t, respectively. In the first period, the parent is 60 years old, i.e., age P 1 = 60. The model incorporates uncertainties about the parent s and the child s preference shocks, and the parent s health and wealth shocks. The parent s health status, denoted by h P t, can take four values; the parent can be healthy (h P t = 0), have light long-term care needs (h P t = 1), have severe long-term care needs (h P t = 2), or be dead (h P t = 3). At the beginning of each period, the parent s health and wealth shocks are realized, and are commonly observed by the parent and the child. If the parent is alive, then the child privately observes the realization of her preference shocks, and moves first by choosing the discrete vector d K t = (ic K t, e K t ) comprising hours of informal care ic K t and labor force participation e K t. The child s flow utility while the parent is alive is represented as: π K (c K t, l K t, ic K t ; h P t, ic K t 1, X K ) + ɛ K t (d K t ) (3) 16 I focus on interactions between one parent and one child because most parents receive informal care from just one child (as discussed in Section 2), and incorporating informal care decisions by spouse and multiple children would result in a substantial computational cost. While the model does not endogenize informal care decisions of multiple children, it incorporates the possibility that the number of children affects the chance of using formal care services in a reduced-form way. Details will be presented in Section

14 where c K t = y K (e K t ; e K t 1, age K t, X K ), (4) l K t = T total ic K t T e K t. (5) The child derives utility from consumption c K t, leisure l K t, informal care provision ic K t, and additive preference shock associated with discrete choice vector d K t, denoted by ɛ K t (d K t ). Appendix B presents the exact specification of utility function π K ; it is additively separable in preferences for consumption, leisure, and informal care hours, and the consumption and leisure preferences follow a constant relative risk aversion utility function. The assumption that informal care hours directly enter the child s utility function implies that the child may be impurely altruistic toward the parent, and may derive utility from the act of providing informal care. This warm-glow utility may also depend on the parent s health status h P t, the child s informal care provision in the previous period ic K t 1, and the child s demographic characteristics XK. The possible dependence upon ic K t 1 is to incorporate costs associated with initiating informal care provision. Equation (4) states that the child s consumption is equal to her income which is determined by her labor force participation choice, past employment, age, and demographic characteristics. Appendix B presents the exact specification of income function y K. The child s leisure time is residually determined by the time constraint in Equation (5) where T total is the child s total endowed time and T e K represents the t required hours for work choice e K t. I assume the preference shock ɛ K t follows an i.i.d. extreme value type I distribution with scale one. In each period, the parent moves after observing the child s choices. This timing assumption is imposed to avoid the issue of multiple equilibria in a simultaneous-move version of the game. While the parent is alive, in each period, the parent makes decisions regarding formal care utilization and consumption. The parent s formal care utilization choice is discrete, and she can use paid home care, enter a nursing home, or not use any formal care. The parent never uses formal care when she is healthy or the child provides informal care. This assumption implies that informal and formal care are substitutes, as found in empirical studies like Charles and Sevak (2005) and Coe, Goda, and Van Houtven (2015). In the first period, the parent is assumed to be healthy (so the parent does not use any formal care), and she chooses whether to buy long-term care insurance once-andfor-all. I consider one standardized long-term care insurance policy with the following features. It covers both paid home care and nursing homes, pays benefits for formal care expenses only when the parent is unhealthy, has a maximal per-period benefit cap, and provides coverage for life. 17 The per-period premium is p(x P, X K ), implying that I allow the premium to vary by the parent and the child s demographic characteristics, X P and X K, respectively. The premium is paid in every period when the parent is not receiving benefits from the insurance company. 17 These features are based on typical long-term care insurance products sold during my sample period (Brown and Finkelstein, 2007; Broker World, ). 14

15 The parent s flow utility when the parent is alive is represented as π P (c P t, fc P t, ic K t ; h P t, n P ) + ɛ P t (d P t ). (6) The parent derives utility from consumption c P t, formal care fc P t, informal care hours chosen and provided by the child ic K t, and additive preference shock associated with discrete choice d P t, denoted by ɛ P t (d P t ). 18 The discrete choice d P t represents the insurance choice in the first period, and formal care choice for all other periods. Appendix B presents the exact specification for utility function π P ; it is additively separable in preferences for consumption, formal care, and informal care, and the consumption preference follows a constant relative risk aversion utility function. The assumption that fc P t and ic K t directly enter the parent s utility function implies that the parent has a preference over formal and informal care, which may depend on the parent s health status. The parent s preference for formal care may also depend on the parent s number of children, denoted by n P. This assumption is to capture the effect of having multiple children on formal care expenditure risk in a reduced-form way. I assume the preference shock ɛ P t is privately observed by the parent before she makes the discrete choice, and follows an i.i.d. extreme value type I distribution with scale one. I now describe preferences when the parent dies. The model closes when the parent s health status is realized to be death, and the parent dies for sure at age 100. The parent leaves her remaining wealth wt P to the family and derives the following bequest utility: πd P (wt P ) = θd P wt P. (7) This linear functional form follows Hurd (1989), Kopczuk and Lupton (2007), and Lockwood (2016), and implies that bequests are luxury goods and the parent is less risk-averse over bequests than over consumption. The child inherits a share of the parent s wealth, and this is how the model incorporates the child s strategic incentive. As the parent does not incur any formal care expenses while receiving informal care from the child, the child may be strategically motivated to provide informal care in order to preserve the parent s savings. The child s terminal value is represented as the following: πd K (wt P, X K ) (8) which depends on the parent s bequest and the child s demographic characteristics. specification for π K d is found in Appendix B. The exact State variables and transition. The set of state variables that are commonly observed by the parent and the child at the beginning of period t, after the resolution of uncertainty about the 18 The parent s flow utility is not a function of leisure because I assume that (1) the parent spends all her time on leisure, and (2) her leisure utility is additively separable. 15

16 parent s health and wealth is: s t = (age P t, h P t, w P t, ltci P t, age K t, ic K t 1, e K t 1; X P, X K ). w P t is the parent s wealth after the wealth shock, and ltci P t is an indicator that is equal to one if the parent has long-term care insurance, and zero otherwise. The value of ltci P t is determined in the first period as a result of the parent s long-term care insurance decision. X P is the set of the parent s demographic characteristics including gender, permanent income, and number of children, and X K represents the child s demographic characteristics including gender, education, marital status, home ownership, and residential proximity to the parent. All variables in s t evolve deterministically except for the parent s health and wealth. The parent s health transition probabilities follow a Markov chain and depend on the parent s gender, age, and current health status. 19 To specify the parent s wealth accumulation law, I first describe how the model incorporates Medicaid. To be Medicaid eligible, the parent s net resources after paying the out-of-pocket cost of formal care must be less than the Medicaid threshold: w P t ( + y P x fc P t,h P t min{b, x fc P t,h P t } ) w fc P t (9) where y P is the parent s permanent income, x fc P t,h P is the price of formal care choice fc P t t in health status h P t, b is the per-period benefit from long-term care insurance (zero if the parent is not an insurance owner), and w fc P is the Medicaid threshold which depends on the parent s formal care t choice. 20 If the parent is Medicaid eligible, then her out-of-pocket cost of formal care is reduced to max{0, w P t + y P w fc P t } and Medicaid pays the remaining cost: x fc P t,h P t min{b, x fc P t,h P t } max{0, wp t + y P w fc P t }. Note that (1) Medicaid is a secondary payer in the sense that long-term care insurance must pay the benefits first, and (2) the parent becomes Medicaid eligible only after having spent down her net resources to the Medicaid threshold. The parent s wealth after paying the long-term care insurance premium and the out-of-pocket cost of formal care, if any, is w t P wt P + y P max{0, wt P + y P w fc P } = min{w = t t P + y P, w fc P } if Medicaid eligible, ( ) t wt P + y P x fc P t,h P min{b, x t fc P t,h P } p(x P, X K ) otherwise. t If the parent does not own any long-term care insurance, then b = p(x P, X K ) = 0. To make sure that the parent maintains strictly positive consumption, there is a government transfer up to g fc P t, 19 This suggests that the parent s health transition process is exogenous and does not depend on the receipt of informal or formal care. This is based on previous studies that find that the evolution of long-term care needs is largely unaffected by the use of long-term care (Byrne, Goeree, Hiedemann, and Stern, 2009). 20 The Medicaid resource threshold for paid home care is substantially higher than that for nursing home care. The modal income threshold for paid home care was $545 per month, while it was only $30 per month for nursing homes in 1999 (Brown and Finkelstein, 2008). (10) 16

17 which depends on the parent s formal care choice. This can be thought of as the Supplemental Security Income (SSI) benefits, which vary by beneficiaries nursing home residency. The parent s wealth after this government transfer is ŵ P t (s t, d P t ) := max{ w P t, g fc P t }. (11) There is no borrowing and the parent s consumption is constrained by c P t ŵ P t (s t, d P t ). The parent s wealth at the beginning of the next period is given by { ( ) } wt+1 P = max 0, (1 + r) ŵt P (s t, d P t ) c P t m P t+1 where r is the real per-period interest rate, and m P t+1 is the i.i.d. wealth shock realized at the beginning of the next period for which the parent is liable up to ŵt P (s t, d P t ) c P t. (12) 3.2 Equilibrium of the Intergenerational Game To define equilibrium decision rules of the family, I first formally define a strategy profile σ = (σ K, σ P ) comprising a set of decision rules for the child σ K and a set of decision rules for the parent σ P. σ K specifies the child s informal care supply and labor force participation decision rules over the parent s life-cycle. Specifically, σ K = {σ K (s t, ɛ K t )} is a mapping from the common state space, S, and the space of the child s private preference shocks, R CK, to the set of the child s informal care and labor force participation choices, C K : σ K : S R CK C K. σ P = (σ P,d, σ P,c ) is composed of the parent s discrete choice decision rules σ P,d and consumption decision rules σ P,c over the parent s life-cycle. In the first period, the discrete choice is the onceand-for-all long-term care insurance purchase choice, and for all other periods, it is the formal care utilization choice. As the parent makes the discrete choice after observing the child s choice, σ P,d = {σ P,d (s t, d K t, ɛ P t )} is a mapping from the common state space, the child s choice set, and the space of the parent s private preference shocks, R CP, to the parent s discrete choice set, C P : σ P,d : S C K R CP C P. The parent chooses consumption after her discrete choice. So σ P,c = {σ P,c (s t, d K t, d P t )} is a mapping from the common state space, the child s choice set, and the parent s set of discrete choices to the strictly positive real line: 21 σ P,c : S C K C P R As the parent s preference shocks (ɛ P t ) are additively separable and serially independent, conditional on the parent s discrete choices, these shocks are irrelevant to consumption choices. 17

18 Let Ṽ K (s t, ɛ K t ; σ) denote the child s value in the state s t after the realization of her private preference shocks ɛ K t if she behaves optimally today and in the future when the parent behaves according to her decision rules specified in σ. In states where the parent is dead, with a slight abuse of notation, define Ṽ K = πd K where πk d is the child s terminal value as defined in Equation (8). In each period while the parent is alive, the child solves the following problem: Ṽ K (s t, ɛ K t ; σ) = { max π K + ɛ K d K t (d K t ) + βe t CK (s t) [Ṽ K (s t+1, ɛ K t+1; σ) s t, d K t ; σ] } (13) where π K is defined as in Equation (3) but all arguments are suppressed for notational simplicity, β is the discount factor, and the expectation is over the parent s private preference shocks of the current period, the parent s health and wealth shocks of the next period, and the child s private preference shocks of the next period. C K (s t ) denotes the set of the child s feasible informal care and labor force participation choices in state s t. Define V K (s t ; σ) as the expected value function, V K (s t ; σ) = Ṽ K (s t, ɛ K t ; σ)g(ɛ K t ) where g is the probability density function of ɛ K t. Define the choice-specific value function, v K (s t, d K t ; σ), as the per-period payoff of choosing d K t preference shock plus the expected value function: [ ] v K (s t, d K t ; σ) = π K + βe V K (s t+1 ; σ) s t, d K t ; σ where again, π K is defined as in Equation (3). minus the I similarly define value functions for the parent. Let Ṽ P (s t, d K t, ɛ P t ; σ) denote the parent s value if the parent behaves optimally today and in the future when the child behaves according to her decision rules specified in σ. Again, with a slight abuse of notation, define Ṽ P = πd P in states where the parent is dead, where πd P is the parent s bequest utility as defined in Equation (7). The parent s problem when she is alive can be written as Ṽ P (s t, d K t, ɛ P t ; σ) = (14) { max π P (c P d P t CP (s t,d K t ),cp t (0,ŵP t (st,dp t )] t ) + ɛ P t (d P t ) + βe [Ṽ P (s t+1, d K t+1, ɛ P t+1; σ) s t, d K t, d P t, c P t ; σ] } (15) where π P (c P t ) is defined as in Equation (6) but arguments other than c P t are suppressed for notational simplicity, the expectation is over the parent s wealth, health, and preference shocks of the next period, and the child s private preference shocks of the next period. C P (s t, d K t ) denotes the set of the parent s feasible discrete choices in state s t when the child s choice is d K t. 22 As there is no borrowing, consumption cannot be greater than the wealth after the government transfer, ŵt P (s t, d P t ) as defined in Equation (11). I define the parent s expected value function as V P (s t, d K ; σ) = Ṽ P (s t, d K t, ɛ P t ; σ)g(ɛ P t ). I denote the parent s choice-specific value function as v P (s t, d K t, d P t ; σ), and it is defined as the parent s per-period payoff of choosing discrete choice d P t 22 The dependence upon d K t is because of the assumption that the parent does not use formal care when the child provides informal care. 18